Finite Element Method for a Conformable Differential Equation
DOI:
https://doi.org/10.19136/jobs.a12n34.6689Keywords:
Non-integer order calculus, Conformable derivative, Calculus of variations, Finite element method, Conformable differential equationsAbstract
This paper presents a numerical scheme for a one dimensional differential equation involving the conformable derivative with Dirichlet boundary conditions, developed within the framework of the finite element method. Starting from a weak formulation based on a conformable version of the fundamental lemma of the calculus of variations, the method is implemented using piecewise-defined test functions over a partition of the domain. The locality of the conformable operator allows for a direct adaptation of standard finite element techniques. Numerical experiments are carried out for different values of the order of derivation α, the results show a smooth transition in the approximated solutions as α approaches one, consistently recovering the classical case.
References
[1] T. Abdeljawad. On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279:57–66, 5 2015.
[2] A. A. Alderremy, H. I. Abdel-Gawad, K. M. Saad, and E. M. Aly Shaban. New exact solutions of time conformable fractional Klein Kramer equation. Optical and Quantum Electronics, 53(12):693, dec 2021.
[3] A. Arafa. A different approach for conformable fractional biochemical reaction—diffusion models. Applied Mathematics-A Journal of Chinese Universities, 35(4):452–467, oct 2020.
[4] J.A. Burns. Introduction to the Calculus of Variations and Control with Modern Applications. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science. CRC Press, 2013.
[5] J. M. Cruz-Duarte, J. J. Rosales-Garc´ıa, and C. R. Correa-Cely. Entropy Generation in a Mass-Spring-Damper System Using a Conformable Model. Symmetry, 12(3):395, mar 2020.
[6] K. Diethelm. The Analysis of Fractional Differential Equations, volume 2004 of Lecture Notesin Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.
[7] A. R. Hadhoud, F. E. I. Abd Alaal, and T. Radwan. Modified finite element numerical method for solving conformable space-time fractional nonlinear partial differential equations. Fractals, 30(10):2240247, 2022.
[8] I. Haouam. On the conformable fractional derivative and its applications in physics. Journal of Theoretical and Applied Physics, 18(6), 2024.
[9] B. B. Iskender and D. Yapı¸skan. Generalized conformable variational calculus and optimal control problems with variable terminal conditions. AIMS Mathematics, 5(2):1105–1126, 2020.
[10] R. Kaviya and P. Muthukumar. Dynamical analysis and optimal harvesting of conformable fractional prey–predator system with predator immigration. The European Physical Journal Plus, 136(5):542, may 2021.
[11] R. Khalil, M. Al Horani, R. Yousef, and M. Sababheh. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264:65–70, 7 2014.
[12] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier B.V., Amsterdam, 2006.
[13] M. J. Lazo and D. F.M. Torres. Variational calculus with conformable fractional derivatives. IEEE/CAA Journal of Automatica Sinica, 4:340–352, 4 2017.
[14] C. Li and F. Zeng. Numerical Methods for Fractional Calculus. Chapman & Hall/CRC Numerical Analysis and Scientific Computing. CRC Press, 2015.
[15] W. K. Liu, S. Li, and H. S. Park. Eighty Years of the Finite Element Method: Birth, Evolution, and Future. Archives of Computational Methods in Engineering, 29(6):4431–4453, oct 2022.
[16] L. Mart´ınez, J.J. Rosales, C.A. Carre˜no, and J.M. Lozano. Electrical circuits described by fractional conformable derivative. International Journal of Circuit Theory and Applications, 46(5):1091–1100, may 2018.
[17] A. Martynyuk, G. Stamov, I. Stamova, and E. Gospodinova. Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis. Mathematics, 11(10):2221, may 2023.
[18] K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons Inc., New York, 1993.
[19] S. Ogrecki and S. Asliy¨uce. Fractional variational problems on conformable calculus. Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 70(2):719–730, dec 2021.
[20] J. Rosales-Garc´ıa, J. A. Andrade-Lucio, and O. Shulika. Conformable derivative applied to experimental Newton’s law of cooling. Revista Mexicana de F´ısica, 66(2 Mar-Apr):224–227, mar 2020.
[21] L. Sadek, T. A. Lazˇar, and I. Hashim. Conformable finite element method for conformable fractional partial differential equations. AIMS Mathematics, 8(12):28858–28877, 2023.
[22] G. Sales Teodoro, J. A. Tenreiro Machado, and E. Capelas de Oliveira. A review of definitions of fractional derivatives and other operators. Journal of Computational Physics, 388:195–208, 7 2019.
[23] K. Soni and A. K. Sinha. Dynamics of epidemic model with conformable fractional derivative. Nonlinear Science, 4:100040, sep 2025.
[24] S¸. Toprakseven. Numerical Solutions of Conformable Fractional Differential Equations by Taylor and Finite Difference Methods. S¨uleyman Demirel ¨ Universitesi Fen Bilimleri Enstit¨us¨u Dergisi, 23(3):850–863, dec 2019.
[25] S. A. Zanib, T. Zubair, S. Ramzan, M. B. Riaz, M. I. Asjad, and M. Taseer. A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease. PLOS ONE, 19(10):e0307707, oct 2024.
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